Universal property of skew PBW extensions
In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examp...
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irk-123456789-1547572019-06-16T01:29:07Z Universal property of skew PBW extensions Acosta, J,.P. Lezama, O. In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others. As a corollary we will give a new short proof of the Poincar\'{e}-Birkhoff-Witt theorem about the bases of enveloping algebras of finite-dimensional Lie algebras. 2015 Article Universal property of skew PBW extensions / J,.P. Acosta, O. Lezama // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 1-12 . — Бібліогр.: 10 назв. — англ. 1726-3255 2010 MSC:Primary: 16S10, 16S80; Secondary: 16S30, 16S36. http://dspace.nbuv.gov.ua/handle/123456789/154757 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others. As a corollary we will give a new short proof of the Poincar\'{e}-Birkhoff-Witt theorem about the bases of enveloping algebras of finite-dimensional Lie algebras. |
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Acosta, J,.P. Lezama, O. |
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Acosta, J,.P. Lezama, O. Universal property of skew PBW extensions Algebra and Discrete Mathematics |
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Acosta, J,.P. Lezama, O. |
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Acosta, J,.P. |
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Universal property of skew PBW extensions |
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Universal property of skew PBW extensions |
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Universal property of skew PBW extensions |
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Universal property of skew PBW extensions |
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Universal property of skew PBW extensions |
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universal property of skew pbw extensions |
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Інститут прикладної математики і механіки НАН України |
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Universal property of skew PBW extensions / J,.P. Acosta, O. Lezama // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 1-12 . — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT acostajp universalpropertyofskewpbwextensions AT lezamao universalpropertyofskewpbwextensions |
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2025-07-14T06:15:24Z |
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2025-07-14T06:15:24Z |
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1837601889778139136 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 1, pp. 1–12
© Journal “Algebra and Discrete Mathematics”
Universal property of skew P BW extensions
Juan Pablo Acosta and Oswaldo Lezama
Communicated by V. A. Artamonov
Abstract. In this paper we prove the universal property of
skew PBW extensions generalizing this way the well known univer-
sal property of skew polynomial rings. For this, we will show first a
result about the existence of this class of non-commutative rings.
Skew PBW extensions include as particular examples Weyl alge-
bras, enveloping algebras of finite-dimensional Lie algebras (and its
quantization), Artamonov quantum polynomials, diffusion algebras,
Manin algebra of quantum matrices, among many others. As a corol-
lary we will give a new short proof of the Poincaré-Birkhoff-Witt
theorem about the bases of enveloping algebras of finite-dimensional
Lie algebras.
1. Introduction
Most of constructions in algebra are characterized by universal prop-
erties from which it is easy to prove important results about the con-
structed object. This is the case of the universal property of the tensor
product; another well known example is the universal property for the
localization of rings and modules by multiplicative subsets. A key exam-
ple in non-commutative algebra is the skew polynomial ring R[x; σ, δ];
the universal property in this case says that if B is a ring with a ring
homomorphism ϕ : R → B and in B there exists and element y such
that yϕ(r) = ϕ(σ(r))y + ϕ(δ(r)) for every r ∈ R, then there exists an
2010 MSC: Primary: 16S10, 16S80; Secondary: 16S30, 16S36.
Key words and phrases: skew polynomial rings, skew P BW extensions, P BW
bases, quantum algebras.
2 Universal property of skew PBW extensions
unique ring homomorphism ϕ̃ : R[x; σ, δ] → B such that ϕ̃(x) = y and
ϕ̃(r) = ϕ(r) (see [9]). In this paper we prove the universal property of
skew PBW extensions generalizing the universal property of skew poly-
nomial rings. For this, we will prove first a theorem about the existence
of skew PBW extensions similar to the corresponding result on skew
polynomial rings. As application we will get the Poincaré-Birkhoff-Witt
theorem about the bases of enveloping algebras of finite-dimensional Lie
algebras. This famous theorem says that if K is a field and G is a finite-
dimensional Lie algebra with K-basis {y1, . . . , yn}, then a K-basis of the
universal enveloping algebra U(G) is the set of monomials yα1
1 · · · yαn
n ,
αi > 0, 1 6 i 6 n (see [4], [6]).
Skew PBW extensions were defined firstly in [7], and their homological
and ring-theoretic properties have been studied in the last years (see [1],
[3], [8], [10]). Skew polynomial rings of injective type, Weyl algebras, en-
veloping algebras of finite-dimensional Lie algebras (and its quantization),
Artamonov quantum polynomials, diffusion algebras, Manin algebra of
quantum matrices, are particular examples of skew PBW extensions (see
[8]). In this first section we recall the definition of skew PBW extensions
and some very basic properties needed for the proof of the main theorem.
Definition 1.1. Let R and A be rings. We say that A is a skew PBW
extension of R (also called a σ − PBW extension of R) if the following
conditions hold:
(i) R ⊆ A.
(ii) There exist finite elements x1, . . . , xn ∈ A such A is a left R-free
module with basis
Mon(A) := {xα = xα1
1 · · · xαn
n | α = (α1, . . . , αn) ∈ N
n}.
In this case it says also that A is a left polynomial ring over R
with respect to {x1, . . . , xn} and Mon(A) is the set of standard
monomials of A. Moreover, x0
1 · · · x0
n := 1 ∈ Mon(A).
(iii) For every 1 6 i 6 n and r ∈ R − {0} there exists ci,r ∈ R − {0}
such that
xir − ci,rxi ∈ R. (1.1)
(iv) For every 1 6 i, j 6 n there exists ci,j ∈ R − {0} such that
xjxi − ci,jxixj ∈ R + Rx1 + · · · + Rxn. (1.2)
Under these conditions we will write A := σ(R)〈x1, . . . , xn〉.
J. P. Acosta, O. Lezama 3
The following proposition justifies the notation and the alternative
name given for the skew PBW extensions.
Proposition 1.2. Let A be a skew PBW extension of R. Then, for every
1 6 i 6 n, there exists an injective ring endomorphism σi : R → R and a
σi-derivation δi : R → R such that
xir = σi(r)xi + δi(r),
for each r ∈ R.
Proof. See [7], Proposition 3.
Observe that if σ is an injective endomorphism of the ring R and δ is
a σ-derivation, then the skew polynomial ring R[x; σ, δ] is a trivial skew
PBW extension in only one variable, σ(R)〈x〉.
Some extra notation will be used in the rest of the paper.
Definition 1.3. Let A be a skew PBW extension of R with endomor-
phisms σi, 1 6 i 6 n, as in Proposition 1.2.
(i) For α = (α1, . . . , αn) ∈ N
n, σα := σα1
1 · · · σαn
n , |α| := α1 + · · · + αn.
If β = (β1, . . . , βn) ∈ N
n, then α + β := (α1 + β1, . . . , αn + βn).
(ii) For X = xα ∈ Mon(A), exp(X) := α and deg(X) := |α|.
(iii) If f = c1X1 + · · · + ctXt, with Xi ∈ Mon(A) and ci ∈ R − {0}, then
deg(f) := max{deg(Xi)}
t
i=1.
The skew PBW extensions can be characterized in a similar way as
was done in [5] for PBW rings.
Theorem 1.4. Let A be a left polynomial ring over R w.r.t. {x1, . . . , xn}.
A is a skew PBW extension of R if and only if the following conditions
hold:
(a) For every xα ∈ Mon(A) and every 0 6= r ∈ R there exist unique
elements rα := σα(r) ∈ R − {0} and pα,r ∈ A such that
xαr = rαxα + pα,r, (1.3)
where pα,r = 0 or deg(pα,r) < |α| if pα,r 6= 0. Moreover, if r is left
invertible, then rα is left invertible.
4 Universal property of skew PBW extensions
(b) For every xα, xβ ∈ Mon(A) there exist unique elements cα,β ∈ R
and pα,β ∈ A such that
xαxβ = cα,βxα+β + pα,β , (1.4)
where cα,β is left invertible, pα,β = 0 or deg(pα,β) < |α + β| if
pα,β 6= 0.
Proof. See [7], Theorem 7.
2. Existence theorem for skew PBW extensions
If A = σ(R)〈x1, . . . , xn〉 is a skew PBW extension of the ring R, then
as was observed in the previous section, A induces unique endomorphisms
σi : R → R and σi-derivations δi : R → R, 1 6 i 6 n. Moreover, by
(1.2), there exist cij , dij , a
(k)
ij ∈ R such that xjxi = cijxixj + a
(1)
ij x1 +
· · · + a
(n)
ij xn + dij , with 1 6 i, j 6 n. However, note that if i < j, since
Mon(A) is a R-basis, then 1 = cj,ici,j , i.e., for every 1 6 i < j 6 n, cji
is a right inverse of ci,j univocally determined. In a similar way, we can
check that a
(k)
ji = −cjia
(k)
ij , dji = −cjidij . Thus, given A there exist unique
parameters cij , dij , a
(k)
ij ∈ R such that
xjxi = cijxixj +a
(1)
ij x1+· · ·+a
(n)
ij xn+dij , for every 1 6 i < j 6 n. (2.1)
Definition 2.1. Let A = σ(R)〈x1, . . . , xn〉 be a skew PBW extension.
σi, δi, cij , dij , a
(k)
ij , 1 6 i < j 6 n, defined as before, are called the param-
eters of A.
Conversely, given a ring R and parameters σi, δi, cij , dij , a
(k)
ij , 1 6 i <
j 6 n, we will construct in this section a skew PBW extension with
coefficient ring R and satisfying the following equations
1) For i < j in I and k in I, xjxi = cijxixj + Σka
(k)
ij xk + dij ,
2) For i ∈ I and r ∈ R, xir = σi(r)xi + δi(r),
where I := {1, . . . , n}.
Definition 2.2. Let R be a ring and W be the free monoid in the alphabet
X ∪ R, with X := {xi : i ∈ I}. Let w be a word of W , the complexity
of w, denoted c(w), is a triple of nonnegative integers (a, b, c), where a
is the number of x’s in w, b is is the number of inversions involving only
x’s, and c is the number of inversions of the type (xi, r).
J. P. Acosta, O. Lezama 5
These triples are ordered with the lexicographic order, i.e., (a, b, c) 6
(d, e, f) if and only if a < d, or, a = d and b < e, or, a = d, b = e and
c 6 f . This is a well order. Let T be the set of elements of W such that
c(w) = (a, 0, 0) and ZT be the linear extension of T in Z〈X ∪ R〉 (the
Z-free algebra in the alphabet X ∪ R).
Definition 2.3. Let R be a ring, {cij}i<j , {dij}i<j and {a
(k)
ij }i<j,k be
elements of R indexed by i, j, k in I. Let σi, δi : R → R be two functions
for each i ∈ I. Suppose that cij is left invertible and that σi(r) 6= 0 for
r 6= 0. We define the function p
p : W → Z〈X ∪ R〉, with X := {xi : i ∈ I},
by induction in the complexity, as follows:
1) If w ∈ T then p(w) = w.
2) If w = v1xirv2, with r ∈ R, v1 ∈ W and rv2 ∈ T then
p(w) = p(v1σi(r)xiv2) + p(v1δi(r)v2).
3) If w = v1xjxiv2, where v1 ∈ W , xiv2 ∈ T with i < j, then
p(w) = p(v1cijxixjv2) + Σkp(v1a
(k)
ij xkv2) + p(v1dijv2).
The linear extension of p to Z〈X∪R〉 → Z〈X∪R〉 is also denoted p. The im-
age of p is contained in ZT . Let Mon := {Πn
k=1xik
: i1 6 · · · 6 in, n > 0},
and FR(Mon) be the left free R−module with basis Mon. We define
q : ZT → FR(Mon) as the bilinear extension of q(r1 . . . rmxi1
. . . xin
) :=
(Πm
k=1rk)xi1
. . . xin
. Finally, we define h : Z〈X ∪ R〉 → FR(Mon) as
h := qp.
Theorem 2.4 (Existence). Let R, I, X, ak
ij , cij , σi, δi, h, p, q be as in Defi-
nition 2.3. Then, there exists a skew PBW extension A of R with variables
X := {xi : i ∈ I} such that
(a) xir = σi(r)xi + δi(r).
(b) xjxi = cijxixj + Σka
(k)
ij xk + dij , for i < j in I.
if and only if
(1) For every i in I, σi is a ring endomorphism of R and δi is σi-
derivation.
6 Universal property of skew PBW extensions
(2) h(xjxir) = h(p(xjxi)r), for i < j in I and r ∈ R.
(3) h(xkxjxi) = h(p(xkxj)xi), for i < j < k in I.
Proof. (=⇒) Numeral (1) is the content of Proposition 1.2. Conditions (2)
and (3) follow from (a) and (b) and the associativity xj(xir) = (xjxi)r
and xk(xjxi) = (xkxj)xi.
(⇐=) Define t : FR(Mon) → Z〈X∪R〉 as t(Σrx̄x̄) := Σrx̄x̄ ∈ Z〈X∪R〉,
where Σrx̄x̄ is the unique expression of an element in FR(Mon) as a sum
over a finite set, x̄ ∈ Mon and rx̄ 6= 0 is an element of R.
We define a product in FR(Mon) by
f ⋆ g = h(t(f)t(g)), f, g ∈ FR(Mon),
and we will prove in Lemma 2.8 below that h(ab) = h(a) ⋆ h(b), with
a, b ∈ Z〈X ∪ R〉. From this we get that h : Z〈X ∪ R〉 → FR(Mon) is
a surjection that preserves sums, products and h(1) = 1. This makes
FR(Mon) a ring, which is a skew PBW extension of R by the definition
of the product ⋆.
To complete the proof we proceed to prove Lemma 2.8, but for this,
we have to show first some preliminary propositions under the hypothesis
(1)-(3).
Proposition 2.5. For a, b ∈ W and r, s ∈ R the following equalities
hold:
(i) h(a0b) = 0.
(ii) h(a(−r)b) = −h(arb).
(iii) h(a(r + s)b) = h(arb + asb).
(iv) h(a1b) = h(ab).
(v) h(a(rs)b) = h(arsb).
Proof. (i) and (ii) follow from (iii) since r 7→ h(arb) is a group homomor-
phism from the additive group of R into FR(Mon).
(iii) is proven by induction on c(a(r+s)b) and applying the definition of
h. Here the conditions δi(a+b) = δi(a)+δi(b) and σi(a+b) = σi(a)+σi(b)
in the hipothesis (1) of Theorem 2.4 are used.
(iv) is proven by induction on c(a1b) and making use of part (i). The
relevant hypothesis are σi(1) = 1 and δi(1) = 0 which are part of the
hypothesis (1) in Theorem 2.4.
(v) This part is proven by induction on c(a(rs)b) and making use
of (iii). The relevant hypothesis are σi(ab) = σi(a)σi(b) and δi(ab) =
σi(a)δi(b) + δi(a)b.
J. P. Acosta, O. Lezama 7
Proposition 2.6. Let y, z ∈ Z〈X ∪ R〉 and a ∈ ZT . Then h(yaz) =
h(ytq(a)z).
Proof. This is because we can obtain tq(a) from a with a finite number
of operations described in Proposition 2.5. Indeed if a ∈ ZT then by
definition of T we heave a = Σnuu where the sum is over u ∈ T , nu ∈ Z
and u = r1,u . . . rm,uxj1
. . . xjk
(j1, . . . jk and m, k depend on u) here
rs ∈ R and 1 6 j1 6 · · · 6 jk 6 n. Then by definition of t, q we
have tq(a) = Σx∈Aa(x)x where A = {x ∈ Mon(X) : a(x) 6= 0}, and
a(x) = Σu∈B(x)nuΠsrs,u ∈ R where B(x) = {u ∈ T : xj1
. . . xjk
= x}.
Using the Proposition 2.5 (i) we obtain that
h(ytq(a)z) = h(yΣx∈Mon(X)a(x)xz).
Using that h is linear we get
h(yΣx∈Mon(X)a(x)xz) = Σx∈Mon(x)h(ya(x)xz).
Using Proposition 2.5 (i),(ii),(iii) we get that
h(ya(x)xz) = Σu∈B(x)nuh(y(Πsrs,u)xz).
Further, using Proposition 2.5 (iv)(v) we get that
h(y(Πsrs,u)xz) = h(yr1,u . . . rm,uxz) = h(yuz).
Proposition 2.7. If x, y, z ∈ Z〈X ∪ R〉 then h(xp(y)z) = h(xyz).
Proof. The identity is linear in x, y, z, so we may assume they are words.
Next we proceed by induction on c(xyz). First assume that the first
inversion from right to left in xyz is in y, say y = w1xjsw2 with s = xi
with i < j or s ∈ R, and sw2 ∈ T . Then
h(xyz) = h(xw1p(xjs)w2z) = h(xp(w1p(xjs)w2)z) = h(xp(y)z)
by the definition of p and induction.
Now assume that the first inversion of xyz is not contained in yz, or
xyz ∈ T , in this case y ∈ T and p(y) = y.
Next, assume that the first inversion of xyz is contained in z say
z = w1xjsw2 with sw2 ∈ T and s = xi with i < j or s ∈ R. Then
h(xyz) = h(xyw1p(xjs)w2) = h(xp(y)w1p(xjs)w2) = h(xp(y)z)
by definition of h and induction.
8 Universal property of skew PBW extensions
Now assume that the first inversion of xyz has a part in y and a part
in z, say y = y′xj and z = sz′ with z ∈ T and s = xi with i < j or s ∈ R.
Assume further that the first inversion of y exists and is contained in y′,
say y′ = w1xks′w2 with s′w2 ∈ T an s′ = xi with i < k or s′ ∈ R. Then
h(xyz) = h(xy′p(xjs)z′) = h(xp(y′)p(xjs)z′)
= h(xp(w1p(xks′)w2)p(xjs)z′) = h(xw1p(xks′)w2p(xjs)z′)
= h(xw1p(xks′)w2xjsz′) = h(xp(w1p(xks′)w2xj)sz′)
= h(xp(y)z)
by definition of h and induction applied alternatively. So the last case is
y = y′xkxj with k > j and z = sz′ with s = xi with i < j or s ∈ R and
z ∈ T . In this case
h(xyz) = h(xy′xkp(xjs)z′) = h(xy′p(xkp(xjs))z′)
by definition of h and induction, also observe
h(xy′p(xkp(xjs))z′) = h(xy′p(p(xkxj)s)z′)
because qp(p(xkxj)s) = qp(xkp(xjs) by hipothesis (2) and (3) in Theorem
2.4, and also by Proposition 2.6. Also
h(xy′p(p(xkxj)s)z′) = h(xy′p(xkxj)sz′) = h(xp(y′p(xkxj))z)
by induction applied twice, and h(xp(y′p(xkxj))z) = h(xp(y)z) by defini-
tion of p, as required.
Lemma 2.8. h(ab) = h(a) ⋆ h(b), for a, b ∈ Z〈X ∪ R〉.
Proof. h(a) ⋆ h(b) = h(tqp(a)tqp(b)) = h(p(a)p(b)) = h(ab), the first
equality is from the definition of ⋆, the second equality is from Proposition
2.6 twice and the third equality is Proposition 2.7 twice.
3. The universal property
In this section we will prove the main theorem about the character-
ization of skew PBW extensions by a universal property in a similar
way as this is done for skew polynomial rings. This problem was studied
in [2] where skew PBW extensions were generalized to infinite sets of
generators.
J. P. Acosta, O. Lezama 9
Theorem 3.1 (Main theorem: The universal property).
Let A = σ(R)〈x1, . . . , xn〉 be a skew PBW extension with parameters
σi, δi, cij , dij , a
(k)
ij , 1 6 i, j 6 n. Let B be a ring with homomorphism
ϕ : R → B and elements y1, . . . , yn ∈ B such that
(i) yiϕ(r) = ϕ(σi(r))yi + ϕ(δi(r)), for every r ∈ R.
(ii) yjyi = ϕ(cij)yiyj + ϕ(a
(1)
ij )y1 + · · · + ϕ(a
(n)
ij )yn + dij .
Then, there exists an unique ring homomorphism ϕ̃ : A → B such that
ϕ̃ι = ϕ and ϕ̃(xi) = yi, where ι is the inclusion of R in A.
Proof. Since A is a free R-module with basis Mon(A), we define the
R-homomorphism
ϕ̃ : A → B, r1xα1 + · · · + atx
αt 7→ ϕ(r1)yα1 + · · · + ϕ(at)y
αt ,
where yθ := yθ1
1 · · · yθn
n , with θ := (θ1, . . . , θn) ∈ N
n. Note that ϕ̃(1) = 1.
ϕ̃ is multiplicative: In fact, applying induction on the degree |α + β|
we have
ϕ̃(axαbxβ) = ϕ̃(a[σα(b)xαxβ + pα,bx
β ])
= ϕ̃[aσα(b)[cα,βxα+β + pα,β ] + apα,bx
β]
= ϕ(a)ϕ(σα(b))ϕ(cα,β)yα+β + ϕ(a)ϕ(σα(b))ϕ(pα,β)(y)
+ ϕ(a)ϕ(pα,b)(y)yβ,
where ϕ(pα,β)(y) is the element in B obtained replacing each monomial
xθ in pα,β by yθ and every coefficient c by ϕ(c). In a similar way we
have for ϕ(pα,b)(y) (observe that the degree of each monomial of pα,bx
β
is < |α + β|). On the other hand, applying (i) and (ii) we get
ϕ̃(axα)ϕ̃(bxβ) = ϕ(a)yαϕ(b)yβ
= ϕ(a)[ϕ(σα(b))yα + ϕ(pα,b)(y)]yβ
= ϕ(a)ϕ(σα(b))yαyβ + ϕ(a)ϕ(pα,b)(y)yβ
= ϕ(a)ϕ(σα(b))[ϕ(cα,β)yα+β + ϕ(pα,β)(y)]
+ ϕ(a)ϕ(pα,b)(y)yβ
= ϕ(a)ϕ(σα(b))ϕ(cα,β)yα+β + ϕ(a)ϕ(σα(b))ϕ(pα,β)(y)
+ ϕ(a)ϕ(pα,b)(y)yβ .
It is clear that ϕ̃ι = ϕ and ϕ̃(xi) = yi. Moreover, note that ϕ̃ is the
only ring homomorphism that satisfy these two conditions.
10 Universal property of skew PBW extensions
Corollary 3.2. Let R be a ring and A = σ(R)〈x1, . . . , xn〉 be a skew
PBW extension of R with parameters σi, δi, cij , dij , a
(k)
ij , 1 6 i, j 6 n. Let
B be a ring with homomorphism ϕ : R → B and elements y1, . . . , yn ∈ B
such that the conditions (i)-(ii) in Theorem 3.1 are satisfied with respect
to the system of parameters σi, δi, cij , dij , a
(k)
ij , 1 6 i, j 6 n, of the ring
R. If B satisfies the universal property, then B ∼= A = σ(R)〈x1, . . . , xn〉.
Moreover, the monomials yα1
1 · · · yαn
n , αi > 0, 1 6 i 6 n are a R-basis
of B.
Proof. By the universal property of A there exists ϕ̃ such that ϕ̃ι = ϕ;
by the universal property of B there exists ι̃ such that ι̃ϕ = ι. Note that
ι̃ϕ̃ι = ι and ϕ̃ι̃ϕ = ϕ. The uniqueness gives that ι̃ϕ̃ = iA and ϕ̃ι̃ = iB.
Moreover, in the proof of Theorem 3.1 we observed that ϕ̃ is not only a
ring homomorphism but also a R-homomorphism, whence
ϕ̃(Mon(A)) = {yα1
1 · · · yαn
n |αi > 0, 1 6 i 6 n}
is a R-basis of B.
Corollary 3.3. Let R be a ring and A = σ(R)〈x1, . . . , xn〉 be a skew
PBW extension of R with parameters σi, δi, cij , dij , a
(k)
ij , 1 6 i, j 6 n. Let
B be a ring that satisfies the following conditions with respect to the
system of parameters σi, δi, cij , dij , a
(k)
ij , 1 6 i, j 6 n, of the ring R.
(i) There exists a ring homomorphism ϕ : R → B.
(ii) There exist elements y1, . . . , yn ∈ B such that B is a left free B-
module with basis Mon(y1, . . . , yn), and the product is given by
r · b := ϕ(r)b, r ∈ R, b ∈ B.
(iii) The conditions (i) and (ii) in Theorem 3.1 hold.
Then B ∼= A = σ(R)〈x1, . . . , xn〉.
Proof. According to the universal property of A, there exists a ring
homomorphism ϕ̃ : A → B given by r1xα1 + · · · + atx
αt 7→ ϕ(r1)yα1 +
· · · + ϕ(at)y
αt ; from (ii) we get that ϕ̃ is bijective.
4. The Poincaré-Birkhoff-Witt theorem
Using the results of the previous sections, we will give now a new short
proof of the Poincaré-Birkhoff-Witt theorem about the bases of enveloping
algebras of finite-dimensional Lie algebras. Recall that if K is a field and G
J. P. Acosta, O. Lezama 11
is a Lie algebra with K-basis Y := {y1, . . . , yn}, the enveloping algebra of
G is the associative K-algebra U(G) defined by U(G) = K{y1, . . . , yn}/I,
where K{y1, . . . , yn} is the free K-algebra in the alphabet Y and I the
two-sided ideal generated by all elements of the form yjyi − yiyj − [yj , yi],
1 6 i, j 6 n, where [ , ] is the Lie bracket of G (see [9]).
Theorem 4.1 (Poincaré-Birkhoff-Witt theorem). The standard mono-
mials yα1
1 · · · yαn
n , αi > 0, 1 6 i 6 n, conform a K-basis of U(G).
Proof. For the ring K we consider the following system of variables and
parameters:
X := {x1, . . . , xn}, σi := iK , δi := 0, ci,j := 1, dij := 0,
[xi, xj ] = a
(1)
ij x1 + · · · + a
(n)
ij xn, 1 6 i, j 6 n.
(4.1)
We want to prove that conditions (1)–(3) in Theorem 2.4 hold. Condition
(1) trivially holds. For (2) we have
h(xjxir) = h(xjrxi) = h(rxjxi) = rxixj + r[xj , xi];
h(p(xjxi)r) = h(xixjr) + h([xj , xi]r) = h(xirxj) + r[xj , xi]
= rxixj + r[xj , xi].
Condition (3) of Theorem 2.4 also holds: In fact,
h(p(xkxj)xi) = h(xjxkxi) + h([xk, xj ]xi)
= h(xjxixk) + h(xj [xk, xi]) + h([xk, xj ]xi)
= xixjxk + h([xj , xi]xk) + h(xj [xk, xi]) + h([xk, xj ]xi)
= xixjxk + (h(xk[xj , xi]) + h([[xj , xi], xk])) + (h([xk, xi]xj)
+ h([xj , [xk, xi]])) + (h(xi[xk, xj ]) + h([[xk, xj ], xi]))
= h(xkxjxi) + h([[xj , xi], xk] + [xj , [xk, xi]] + [[xk, xj ], xi])
= h(xkxjxi).
The last equality holds by the Jacobi identity, the second to the last
equality follows regrouping the terms and applying the definition of h to
h(xkxjxi).
From Theorem 2.4 we conclude that there exists a skew PBW exten-
sion A = σ(K)〈x1, . . . , xn〉 that satisfies (4.1), in particular, the monomials
xα1
1 · · · xαn
n , αi > 0, 1 6 i 6 n, conform a K-basis of A. But note that
U(G) satisfies the hypothesis in Corollary 3.2, so U(G) ∼= A and U(G) has
K-basis yα1
1 · · · yαn
n , αi > 0, 1 6 i 6 n.
12 Universal property of skew PBW extensions
References
[1] Acosta, J.P., Chaparro, C., Lezama, O., Ojeda, I., and Venegas, C., Ore and Goldie
theorems for skew PBW extensions, Asian-European Journal of Mathematics, 6
(4), 2013, 1350061-1; 1350061-20.
[2] Acosta, J. P., Ideales Primos en Extensiones P BW torcidas, Tesis de Maestría,
Universidad Nacional de Colombia, Bogotá, 2014.
[3] Chaparro, C., Valuations of skew quantum polynomials, Asian-European Journal
of Mathematics, 8(2), 2015.
[4] Dixmier, J., Enveloping Algebras, GSM 11, AMS, 1996.
[5] Bueso, J., Gómez-Torrecillas, J. and Verschoren, A., Algorithmic Methods in
noncommutative Algebra: Applications to Quantum Groups, Kluwer, 2003.
[6] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, GTM
9, Springer, 1980.
[7] Lezama, O. and Gallego, C., Gröbner bases for ideals of sigma-PBW extensions,
Communications in Algebra, 39 (1), 2011, 50-75.
[8] Lezama, O. & Reyes, M., Some homological properties of skew P BW extensions,
Comm. in Algebra, 42, (2014), 1200-1230.
[9] McConnell, J. and Robson, J., Non-commutative Noetherian Rings, Graduate
Studies in Mathematics, AMS, 2001.
[10] Venegas, C., Automorphisms for skew P BW extensions and skew quantum poly-
nomial rings, Communications in Algebra, 42(5), 2015, 1877-1897.
Contact information
Juan Pablo Acosta,
Oswaldo Lezama
Departamento de Matemáticas
Universidad Nacional de Colombia, Sede Bogotá
E-Mail(s): jolezamas@unal.edu.co
Received by the editors: 02.03.2015
and in final form 16.03.2015.
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